Schedule

 Abstracts

Lidia Angeleri Hügel: Complements.

The existence of complements to large partial tilting objects goes back to work of Riccardo Colpi and Jan Trlifaj in 1995, later revisited together with Tonolo in 2007. In my talk, I will review these old results and illustrate how they inspired recent developments in silting theory. The talk will rely on joint work with Frederik Marks, David Pauksztello and Jorge Vitória.

Silvana Bazzoni: Minimal approximations and Enochs Conjecture.

Enochs Conjecture (or Enochs question) asks if a class of modules providing for covers is necessarily closed under direct limits.

In a recent paper in collaboration with Leonid Positselski and Jan Šťovíček (http://arxiv.org/abs/1911.11720) it is shown that every direct limit presentation has locally split kernel (even quasi split) and this plays a key role in connection with precovers and covers. In particular, this allows to obtain a simple elementary proof of Enochs Conjecture for the left class of an n-tilting cotorsion pair in a Grothendieck category.

The paper deals also with covers and precovers of direct limits in AB3 categories, but in this talk I will mostly deal with module categories and I will discuss some classes of modules over commutative rings providing for minimal approximations, that is covers and/or envelopes.

Simion Breaz: On the cancellation property for a class of mixed groups.

An abelian group G has the cancellation property if whenever H and K are abelian groups such that G⊕H ≅ G⊕K it follows that H≅K. In this talk, I will prove that if a self-small abelian group G of torsion-free rank at most 3 has the cancellation property, then its Walk-endomorphism ring has the unit lifting property. In particular, a strongly indecomposable self-small abelian group G of torsion-free rank at most 3 has the cancellation property if and only if 1 is in the stable range of its Walk-endomorphim ring or there exists a finite group B such that G ≅ B⊕ℤ.

Manuel Cortés Izurdiaga: Tor pairs: products and approximations.

In this talk we will focus on the problem of when, for a given Tor-pair (𝒯, 𝒮) in the categories of modules over a ring R, the class 𝒯 is closed under products or, more generally, products of modules in 𝒯 have finite projective dimension relative to 𝒯. As we will see, this property is related with some Mittag-Leffler conditions satisfied by the modules belonging to 𝒮. As a particular case of our results, we will obtain the characterization of those rings for which the class of all modules with finite flat dimension is closed under products.

This work is motivated by the study of right Gorenstein regular rings, that is, rings with finite right global Gorenstein projective dimension (equivalently, rings for which the classes of right modules with finite projective and with finite injective dimensions coincide).

Alberto Facchini: Noncommutative rings and their spectra.

In Geometry and Topology there are several functors between geometric spaces or topological spaces on the one hand, and commutative rings on the other. The most famous of these functors are probably the ring C(X) of real continuous functions of a topological space X and the spectrum Spec(R) of a commutative ring R. We will discuss the possibility of extending these classical ideas to the noncommutative setting. I will try to speak in a language understandable to everybody (i.e., to any mathematician...)

References:

1. M. Reyes, Obstructing extensions of the functor Spec to noncommutative rings, Israel J. Math. 192 (2012), 667-698.
2. A. Facchini and L. Heidari Zadeh, On a partially ordered set associated to ring morphisms, J. Algebra 535 (2019), 456-479.
3. A. A. Bosi and A. Facchini, A natural fibration for rings, to appear in Rend. Sem. Mat. Univ. Padova, 2020.

Xianhui Fu: Lattice theoretic properties of approximating ideals.

In this talk, it is proved that a finite intersection of special preenveloping ideals in an exact category (𝒜; ℰ) is a special preenveloping ideal. Dually, a finite intersection of special precovering ideals is a special precovering ideal. If the exact category has exact coproducts, resp., exact products, these results extend to infinite families of special peenveloping, resp., special precovering, ideals. This leads to an ideal version of Bongartz' Lemma, which states that if a: A → B is a morphism in 𝒜, then the ideal a^⊥ is special preenveloping. In contrast to the classical version, the Ideal Bongartz' Lemma requires only minimal assumptions, and so may be interpreted as the first step in the Eklof-Trlifaj Lemma. The Ideal Bongartz' Lemma implies that the ideal cotorsion pair generated by a small ideal is complete. This is joint work with I. Herzog, J.S. Hu, and H.Y. Zhu.

Dolors Herbera: The stable category and monoids defined by systems of supports.

Let M be a finitely generated module, and let Add_ℵ0(M) denote the category of the direct summands of M^(ω₀). Let V^*(M) denote the monoid of isomorphism classes of the objects in Add_ℵ0(M).

We will describe a process of "gluing" together the quotients of the category Add_ℵ0(M) by the ideal generated by an object of Add_ℵ0(M), to construct a submonoid B(M) of V^*(M). What makes B(M) particularly interesting, is that in the case of a module finite algebra over commutative noetherian ring, B(M)=V^*(M).

The particular way we found these monoids, motivated us to name them as monoids defined by a system of supports. We will develop some theory about them, mainly in the case of monoids defined by a full system of supports, which is the one appearing when R is a commutative local noetherian ring.

The talk is based on joint work with P. Příhoda and R. Wiegand.

George Ciprian Modoi: Constructing TTF triples in triangulated categories via approximations.

Let 𝒯 be a triangulated category with (co)products and let 𝒞 be a set of compact objects in 𝒯. Then the construction of a t-structure whose aisle is the smallest containing 𝒞 is well–known. We generalize and dualize this construction. Moreover, the main aim of this talk is to combine the direct construction and its dual in order to obtain (co)suspended TTF triples in 𝒞. Some applications are also considered.

Mike Prest: Modules and motives.

To every module one may associate a small abelian category which, under one interpretation, is a category of functors on the definable category generated by that module. I turns out that Nori motives (from algebraic geometry) can be realised as such a category. I will describe this and some associated ideas and constructions.

Sylvia Wiegand: Vanishing of Tor over fiber products.

This is joint work with T. H. Freitas, V. H. Jorge Pérez and R. Wiegand. Let (S,𝔪,k) and (T,𝔫,k) be local rings, and let R denote their fiber product over their common residue field k. Inspired by work of Nasseh and Sather-Wagstaff, we explore consequences of vanishing of Tor^R_m(M,N) for various values of m, where M and N are finitely generated R-modules.